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Geometric Modelling and Deformation for ShapeOptimization of Ship Hulls and Appendages

Elisa Berrini, Bernard Mourrain, Yann Roux, Mathieu Durand, GuillaumeFontaine

To cite this version:Elisa Berrini, Bernard Mourrain, Yann Roux, Mathieu Durand, Guillaume Fontaine. Geomet-ric Modelling and Deformation for Shape Optimization of Ship Hulls and Appendages. Jour-nal of Ship Research, Society of Naval Architects and Marine Engineers, 2017, 61 (2), pp.91-106.�10.5957/JOSR.61.2.160059�. �hal-01373249v4�

Geometric modelling and deformation for shape optimisation of ship hulls and appendages

Elisa Berrinia,b, Bernard Mourraina, Yann Rouxb,c, Mathieu Durandc, Guillaume Fontainec

a Universite Cote d’Azur, INRIA Sophia-Antipolis Mediterranee, 2004 route des Lucioles, 06902 Sophia-Antipolis, Franceb MyCFD, 29 Avenue des freres Roustan, 06220 Golfe-Juan, France

c K-Epsilon, WTC Bat E, 1300 Route des Cretes, 06560 Valbonne, France

Abstract

The precise control of geometric models plays an important role in many domains such as Computer Aided geometric Design andnumerical simulation. For shape optimisation in Computational Fluid Dynamics, the choice of control parameters and the way todeform a shape are critical. In this paper, we describe a skeleton-based representation of shapes adapted for CFD simulation andautomatic shape optimisation. Instead of using the control points of a classical B-spline representation, we control the geometry interms of architectural parameters. We assure valid shapes with a strong shape consistency control. Deformations of the geometryare performed by solving optimisation problems on the skeleton. Finally, a surface reconstruction method is proposed to evaluatethe shape’s performances with CFD solvers. We illustrate the approach on two problems: the foil of an AC45 racing sail boat andthe bulbous bow of a fishing trawler. For each case, we obtained a set of shape deformations and then we evaluated and analysedthe performances of the different shapes with CFD computations.

Keywords: computers in design, hydrodynamics (hull form), design (vessels)

1. Introduction

Automatic shape optimisation is a growing field of study,with applications in various industrial sectors. As the perfor-mance of a flow-exposed object can be obtained accurately withCFD (Computational Fluid Dynamics), small changes in de-sign can be captured and analysed. Based on these performanceanalysis capabilities, optimisation strategies can then be appliedto deform the geometric model in order to improve the physicalbehaviour and the performances of the model.

Fig.1 shows the core of an optimisation loop, which willrepeatedly run numerical simulations and deform the geometryfor an automatic search of an optimal shape.

Figure 1: Automatic shape optimisation loop

Different types of tools need to be linked together to per-form such automatic shape optimisation with aerodynamic orhydrodynamic criteria: a parametric modeller, a meshing tool,a flow solver and an optimisation algorithm [1, 2, 3], see Fig.1.Recent technological progresses allow to quasi-automatically

Email addresses: elisa.berrini@inria.fr / elisa@mycfd.com

(Elisa Berrini), Bernard.Mourrain@inria.fr (Bernard Mourrain),yann@k-epsilon.com (Yann Roux)

run the meshing tool, the CFD solver and the post-processingof the relevant results of the computation. Then optimisationalgorithms such as EGO (Efficient Global optimisation) [4, 5,6] demonstrate their efficiency to solve problems with a largenumber of degrees of freedom and where the objective functionvalues are difficult and costly to evaluate.

However, less efforts have been dedicated to the develop-ment of efficient parametric modellers. These components de-form the object according to the optimisation algorithm output.Their role is critical in the way the space of possible shapes isexplored. To be compatible with the current capabilities of theoptimisation tools, the parametric modeller has to modify theshape of the object using a reduced number of parameters. Itshould provide a precise control of the shape, while allowing togenerate a wide range of admissible shapes.

In this paper, we propose a new approach to shape defor-mation for parametric modellers with the purpose of being in-tegrated into an automatic shape optimisation loop with a CFDsolver.

The methodology presented here has the ability to gener-ate valid shapes from an architectural point of view thanks to anovel shape consistency control based on architectural parame-ters. We focus on reducing the number of degrees of freedomof the deformation problem. We also focus on being indepen-dent from the CAD (Computer Aided-Design) software used byrepresenting objects with a skeleton generated from the CADmodel. Finally, we propose a methodology to link shape repre-sentation, shape deformation and numerical simulation.

Preprint submitted to Journal of Ship Research April 28, 2017

The motivation of working with architectural parameters islead by the intuitiveness for an architect to control a shape bysuch expert parameters instead of control points.

Controlling shapes by architectural parameters allows re-ducing the number of degrees of freedom of the problem. Theyalso introduce a physical and a design meaning into the optimi-sation process, allowing also to generate a majority of shapesthat are valid. In comparison to fixing limits of variation forcoordinates of control points, fixing the bounds to the archi-tectural parameter variation is intuitive. Finally, architecturalparameters are independent of each other, making the search ofan optimal solution easier.

We propose a way to control shapes efficiently in terms ofthese architectural parameters, by controlling and deformingthe skeleton curves in terms of these parameters.

The skeleton deformation is completed by a surface recon-struction step, to produce a smooth geometric model that can beused by the meshing and simulation tools. The approach allowsus to be independent of the initial CAD representation and isnot limited to a specific CAD software.

The generalizable concept of skeleton-based representationis well adapted to extend our tool to a large set a shapes e.g.hulls, appendages, propellers, wind turbine blades, airships.

In this paper, we illustrate application of the modeller ontwo applications: the AC45 foil used by racing yachts, and thebulbous bow of a trawler ship. For each case, we present thechosen hydrodynamic criteria to measure the performances, theshape parameter that we will modify and then we propose ananalysis of the results.

2. Related work

In CAD software, the standard description used to describeshapes are B-Spline curves and surfaces [7]. A B-Spline curveof degree p is defined as :

C(t) =

n∑i=0

Bi,p(t)ci, t ∈ [0, 1] (1)

where ci = (xi, yi, zi) are the 3D control points, and Bi,p(t) arethe B-Spline basis functions.

For CFD computation, the object geometry is representedby a mesh. We present in the following paragraph existingmethods based on both surface or mesh representations of shapes.

Deforming the control points of the NURBS representationsused in CAD software to automatically generate new shapesis not appropriate. The number of control points to representadequately the shape may be too large (3 degrees of freedomper control point) to be used in shape optimisation. Anotherobstacle is the complexity of the geometric models that canbe trimmed, or subdivided into numerous patches that cannotbe deformed in a structured way or that are simply not clean

enough for CFD computations.

Shape deformation of ships forms for automatic shape op-timisation is a relatively recent approach. However, deforma-tion techniques have been highly developed in other applicationfields, such as 3D animation and movies.

Free Form Deformation FFD and morphing are classicalmethods created for 3D animation purposes, and they have beenapplied to shape optimisation for ships. Closely related to theFFD method, deformation techniques that enclose a shape in amesh cage linked with barycentric coordinates have been pro-posed [8]. Naval applications with morphing can be found in[9, 10] and applications with FFD can be found in [11, 12, 13].

FFD and morphing are usually applied to meshes and notto a continuous geometry, thus limiting deformation becausethe meshes can be subject to degeneration. FFD methods canbe very efficient with a small number of degrees of freedom tocontrol the whole shape of the object. However, in order to per-form local deformation, the only way is to increase the numberof control points by refining the areas of interest. Moreover,FFD does not take into account any architectural parameterswhen deforming an object, leading to non-realistic results.

Morphing is limited to known bounds of shape variations.The exploration of the space of possible optimal shapes is ex-tremely reduced.

For 3D animation, another common technique for control-ling shape are skeleton-based mesh deformation techniques [14].We can also find deformation techniques with subdivision sur-faces [15], for example by using energy minimization tech-niques to find the best position of mesh elements to match theuser’s manipulations.

For applications with a direct interaction with the physi-cal characteristics of the object, physically driven deformationmethods exist. In this type of methods, the shape represents thedomain where Partial Differential Equations (PDE) are solved.The domain can be either a mesh or a level set function. The re-sults of the PDE are used in a cost function to determine parts ofthe domain needing to be deformed to optimise its value. Ap-plications can be found for meshes [16], subdivision surfaces[17] and on level set function [18]. Generally in such applica-tions, solving the PDE is not excessively costly. In shipbuild-ing, methods based on shape gradients focuses on minimizingan energy function obtained by solving the Navier-Stokes equa-tions [19, 20].

Engineering dedicated CAD software can also provide para-metric design features, allowing the user to build parametrizedmodels such as CatiaTM or Grasshopper for Rhinoceros 3DTM

or CAESES from Friendship SystemTM. These method allowsto generate shapes easily, but all of the parameters defined onthe design are lost when saving the model in a standard fileexchange format such as IGES or STEP. This represents a limi-tation for automatic linking with solvers (CFD, structural anal-ysis, etc.) or optimisation algorithms.

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These type of parametrized models have been combinedwith isogeometric flow solvers for ship hull optimisation, forinstance in [21, 22].

Specific software have been developed during the last decadesfor ship applications. One of the most widespread is CAE-SES, allowing the user to modify imported geometries usingadvanced geometrical parameters that can be modified by handor automatically with a CFD optimisation loop [23]. The shapecan be controlled with curves or surfaces around the objectcalled Lackenby shift transformations. Points of the object arelinked to the curves or surfaces, and follow its deformations.

Similarly, a ship dedicated tool Bataos [24] allows to mod-ify the shape of sections of the hull by multiplying or addingpredefined functions to the control points of the B-Spline curvedescribing the section.

These tools are based on geometrical control of shapes. Ar-chitectural parameters are computed on the deformed geometryand can be included as constraints, but they do not directly con-trol the shape modification.

3. Shape parametrization

Our goal is to develop a generic methodology to deformshapes with architectural constraints. To achieve this objective,we use a twofold parametrization of the shape that allows us todescribe a large class of objects in the same way. We base ourmethod on a generic skeleton concept to describe the geometry,completed by specific architectural parameters according to thestudied shape.

3.1. Geometrical parametrization

Our motivation for using a skeleton based representation ofthe geometry comes from two considerations:

1. Lines plan are used by naval architects to define the ex-ternal shape of the hull, for which consistent shapes mustbe obtained once deformed.

2. Classical and efficient techniques in 3D animation arebased on the deformation of medial axis curves associ-ated to a shape [14].

By combining these two types of representations, we aim atapplying generic deformation algorithms while controlling thearchitectural consistency.

We consider the skeleton as a set of curves composed of agenerating curve and section curves. Each section curve needsto be identified on the generating curve: a local coordinate sys-tem, with an origin and a rotation, allows us to know its posi-tion and orientation. We are going to describe more preciselythis skeleton based representation in the next section and howthe architectural parameters are associated to the geometry inthe following sections.

3.1.1. Generating curve and section curvesIn our skeleton concept, the generating curve describes the

general shape of the object, whereas sections describe more pre-cisely the outlines of the object around the generating curve,similarly to the architect’s line plan.

The generating curve needs to describe the prominent fea-tures of the object. It is defined to be lying on the geometry andconnects all the section curves. It is not necessarily planar, butsymmetry considerations of the object allow to describe it as aplanar curve in most cases.

Section curves are computed as the intersection curves be-tween the studied object and a family of planes. To each sectioncurve, we associate a point on the generating curve, a local co-ordinate system, an origin and a rotation which allows to knowthe position and the orientation of the section curve. The cut-ting planes are defined to be normal to the tangent vector of thegenerating curve at the corresponding point adjusted with therotation associated to the section.

For practical purposes, we represent the generating curveand the section curves as B-splines curves with a given num-ber of control points. We further simplify the representation bychoosing a finite subset of the section curves, associated with afinite sampling of the generating curve. Fig.2 illustrates skele-tons obtained with Rhinoceros 3DTM.

(a) Skeleton of a sail boat’s foil

(b) Skeleton of a bulbous bow

Figure 2: Examples of skeletons

This leads to a representation of the geometry in terms of afinite set of control points. We denote by cg the control pointsof the generating curve and by ci the control points of the ith

sampled section curve for i = 1, . . . ,N.

We illustrate in the next paragraphs the method to obtain theskeleton on two different models.

To construct a skeleton-based representation from an initialgeometric model, we first choose a relevant generating curve

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according to the model. For airfoil based shapes, the trailingedges is an ideal choice, as is the keeline for a hull.

To obtain the section curves, we compute the intersectionbetween the object and the set of planes defined according tothe tangent of the generating curve. If a non-null rotation isassociated to the section, the cutting plane is first transformedaccording to this rotation. The planes are sampled along thegenerating curve, following a chord length or a curvature baseddistribution. At this stage, we obtained a first skeleton from themodel. The number of control points or the quality (continuity,smoothness, etc.) of the curves depends on the original design.

Then, we reconstruct new B-Spline curves with a fittingprocess [25] from a point cloud sampled on the current sectionsand the generating curve. We use a small number of controlpoints (e.g. ≤ 10) to represent these curves, that are smoothedand cleaned. In the applications that we have considered this isusually enough to ensure a good level of approximation. Theaverage normalized distance between the intersection curvesand the B-spline section curves is kept under 10−5 m.

3.1.2. Local coordinate systems for the section curvesSection curves are identified on the generating curve thanks

to a local coordinate system. Each local coordinate system hasits origin defined from a point on the generating curve, allowingto locate the section in 3D space.

The first axis U is defined into the section plane, its direc-tion is imposed by a main feature of the section, as the leadingedge for an airfoil section, or the maximum height for a bulbousbow section.The second axis V is represented by the tangent of the generat-ing curve Tσ(t) at the origin point of the local coordinate system.In most cases, the first and second axes are orthogonal by con-struction, but some sections representing special features of thegeometry, as the extremities of the foil, are not defined in theplane PT orthogonal to Tσ(t). Thus U and V are not orthogonalto each other.

Let RT be the rotation that transform U such as U ∈ PT . Weapply the inverse of the rotation RT to Tσ(t) in order to obtain thesecond axis V orthogonal to U. RT is associated to the section.

Then the third axis W is computed as the cross product ofthe first two axes.

The implicit definition of the second axis V allows the lo-cal coordinate system to move when the generating curve ismodified, computing the new orientation of the section auto-matically. The translation and rotation matrices that turns theoriginal tangent to the new one is applied to the other axis Uand W and to the section control points. Therefore modifyingthe generating curve induces affine transformations on the sec-tion curves, given by the modification of the local coordinatesystem.

3.2. Architectural parametersArchitectural parameters describe the main characteristics

of the object. They are chosen according to the design practice

and effects on the object performance. Our goal is to control theshape of the studied object through the architectural parametersvalue.

We associate different parameters to the generating curveand the section curves in order to control the whole shape.

For example, the main characteristics of an L-shaped sailboat foil are the length of the two parts, the angle betweenthem and the angle of the entire foil called Cant. Then each air-foil profile section has particular features, such as chord length,thickness, angle of attack, etc. [26].

For a bulbous bow, the main features are the length, theangle, the height and thickness [27]. We illustrate those param-eters in Fig.3 and Fig.4. The generating curve begins where thebulbous bow start to influence the hull shape. The length pa-rameter is the total length of the generating curve. The angleparameter measures the angle between the x-axis and the ex-tremity of the bulb. Variations of these parameters are shown inFig.18. New types of parameters can be implemented easily toenrich the model, such as the sectional areas curve, the volumeof the bulb, etc.

(a) Generating curve parameters

(b) Section parameters (airfoil parameters)

Figure 3: Foil parameters

(a) Generating curve parameters (b) Section pa-rameters

Figure 4: Bulbous bow parameters

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3.3. Observer function

We call φ, the observer function that computes the set ofarchitectural parameters P on a given geometry G: φ : G −→ P.These parameters can be real values such as the length of a foilor functions of the generating curve parameter, such as the twistangle of a profile defined at each point of the generating curve.For a given geometry σ ∈ G, the architectural parameters φ(σ)can thus belong to an infinite dimensional space since it cancontain functions which represents values along the generatingcurve.

In practice, these functions will be represented with a B-Spline curves passing through the section parameter values ac-cording to their position on the generating curve. The B-Splinecurves belong to a finite dimensional space with a small num-ber of control points. These are the parameters that we will useto control the shape.

An illustration is shown in Fig.5, where the observer func-tion made of 13 control points represents the chord length dis-tribution of 28 section curves.

From this consideration, managing the B-Spline instead ofeach section parameters represents two main advantages. First,we reduce drastically the number of parameters that control theshape of the object and that are used in an optimisation loop.Secondly, the modification of a B-Spline curve can ensure asmooth distribution of the parameters, preserving the fairness ofthe object. The observer function can be split into a part for thegenerating curve and a part for the section curves, as differentset of parameters can be defined on each type of curves.

Figure 5: Distribution of the chord length parameter along the generating curveof a foil, approximated with a B-spline curve called observer function

4. Shape deformation

This section explains our strategy for computing a smoothshape corresponding to given architectural parameters. We de-scribed the problem as a non-linear constrained optimisationproblem that can be applied on the generating curve or the sec-tion curves independently.We start by presenting the problem, then we propose an optimi-sation algorithm to solve it numerically.

4.1. Problem setting

Our goal is to find the shape of G that matches a given setof architectural parameters in P.

The observer function φ : G −→ P is defining the param-eters associated to a shape. To control the shape of the objectthrough the parameters value, we need to find a shape corre-sponding to given parameters. In other words, we need to com-pute: φ−1 : P −→ G.

As the shape in G is described by a skeleton made of B-Spline curves, we propose a method that computes new valuesof the coordinates of B-Spline curves control points until thenew skeleton parameters reaches the target ones. The new coor-dinates of the B-Spline control points are the solution of a min-imisation system that we construct with four terms. The discre-tised geometry, represented by a finite number of section curvesand a generating curve is called ξ. The generating curve param-eterized by t ∈ [0, 1] is denoted ξg. Its controlled coefficients arecg. The ith sampled section curve corresponding to the param-eters ti on ξg is denoted ξi for i = 1, . . . ,N. It is parameterizedby s ∈ [0, 1] and its control points are ci = (ci,0, . . . , ci,M). Inthe following paragraphs, ξ0 denotes the initial geometry, thatis the initial generating curve and section curves.

Parameters valueThe first term measures the distance of the current parametersvalues φ(ξ) to the target ones V:

Eparam = ‖φ(ξ) − V‖2 (2)

As we assume that the observer function can be split into apart for the generating curve and a part for the section curves,this error term is the sum of an error term for the generatingcurve and error terms for the sections. We denote by Eparam,i =

‖φi(ξi) − Vi‖2 and Eparam,g = ‖φg(ξg) − Vg‖

2 the error term cor-responding to the ith section curve and the generating curve re-spectively.

Shape consistency controlThe second term is introduced to ensure consistency controlby measuring the distance of the current generating or sectioncurve to the original one. The consistency with the initial geom-etry is measured after applying a linear transformation whichallows to match some parameters of the target curve ξ. Thesetransformations include a scaling of the initial curve to matcha given length or a rotation to match a given angle. In additionto these basics transformations, we also consider other explicitdeformations which depend on the parameters V , such as a non-linear scaling of the height of a profile. These transformationsof the initial curve, denoted DV , are explicitly computed fromthe geometry ξi (or ξg). Transforming the initial curve by DV

helps matching the target parameters. The transformed geome-try DV (ξ0

i ) (or DV (ξ0g)) is used as the starting point of the opti-

misation algorithm.We define:

Eshape,i = ‖ξi − DV (ξ0i )‖2 (3)

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Similarly, for the generating curve we have Eshape,g = ‖ξg −

DV (ξ0g)‖2.

Architectural constraintsThe third term allows taking into account specific constraints Ffor the studied object, usually position or tangency constraints.These constraints are defined for each section ξi, i = 1, . . . ,Nand are not necessarily the same for all sections. For exam-ple, an airfoil has a smooth connection between the suction andpressure faces thanks to a tangency constraint: the tangent atthe leading edge has to be orthogonal to the chord vector:

F1 :∂ξi

∂s·−−−−→chord = 0.

For a bulbous bow, as we parametrized a half bulbous bow, wehave to ensure that the sections end at Y = 0 and that the tangentat the extremity are preserved:

F0 : Y(ξi(1)) = 0 F1 :∂ξi

∂s(v) −

∂ξ0

∂s(v) = 0 v ∈ {0, 1}

RegularizationThe last term controls the overall smoothness of the shape by

introducing stiffness between successive control points ci, j. Weadd correction terms to control respectively C1 and C2 proper-ties of control points.

H0(c) =∑

j

‖∆c j‖2 ∆c j = c j − c j−1

H1(c) =∑

j

‖∆2c j‖2 ∆2c j = c j+1 − 2c j + c j−1

Finally, the proposed minimisation system is described asfollows.

minci

Eparam,i + εEshape,i +∑

k

λkF2k (ci) +

1∑l=0

µl Hl(ci) (4)

As we decoupled it into a minimization system for each sectioncurve ξi and for the generating curve ξg, an optimisation prob-lem similar to Eq.4 is solved for the generating curve ξg.

In these formulations, ε is a weight allowing to balance theinfluence of the shape control term. In fact, if this term is toohigh, the system will converge to a solution too close to theinitial curv, and will have difficulty to respect the target param-eters. ε can be seen as a penalty coefficient, but we chose todecrease it at each iteration because in our particular case theinitial guess, the original curve, needs to be degraded to matchnew architectural parameters. The coefficients λi weighing theshape constraints and µi weighing the correction matrices areboth very small, usually around 10−4.

4.2. Numerical solutionThe definition of the problem is well adapted to Sequential

Quadratic Programming (SQP) [28]. SQP algorithm solves theminimisation problem by generating successive quadratic prob-lems that approximate the cost function by a quadratic functionobtained from a Taylor expansion of order 2. We use finite dif-ference to compute the gradients of the system. We start withan initial value of ε and the original curve as the starting pointof the algorithm, then we decrease ε at each iteration and startthe SQP again with the last computed curve. The algorithmstops when the value of the objective function reaches a fixedthreshold.

Fig.6 illustrates the deformation process. This inner prob-lem, solved by SQP, has a relatively large number of degreesof freedom as it modifies the coordinates of control points. Butwith our methodology based on architectural parameters, theshape optimisation outer loop (see Fig.1) controls only a fewnumber of degrees of freedom.

Figure 6: Deformation process

ExampleFig.2(a) illustrate the skeleton we obtained for the AC45

foil. The original model is made of 22164 control points. Ourskeleton representation is made of only 578 points, 560 = 2 ×28 × 10 points for the sections and 18 points for the generatingcurve.

The associated deformation process can be decoupled into29 independent sub-problems. 28 sub-problems representingthe suction and pressure faces of each sections are controlledby 6 architectural parameters. Each sub-problem has 16 de-grees of freedom, as the extremal points of the sections are pre-determined by chord and angle of attack. The last sub-problem,representing the generating curve, is controlled by 4 architec-tural parameters and has 36 degrees of freedom.

For the bulbous bow, whose skeleton is illustrated in Fig.2(b),the total number of control points for the half-hull is 2574, and57 are directly controlling the bulbous bow shape. Our skeletonrepresentation is made of 180 points, 160 = 16 × 10 points forthe sections and 20 points for the generating curve. We chooseto define more control points on the shape of the bulb than inthe original model, as we look for a precise control of the shapeof the bulbous bow. The number degrees of freedom used by

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the shape optimisation problem is smaller as they are definedby the architectural parameters.

The associated deformation process can be decoupled into11 independent sub-problems. 10 sub-problems representingthe sections are controlled by 2 architectural parameters. Eachsub-problem has 16 degrees of freedom, as the extremal pointsof the sections are pre-determined by the height. The last sub-problem, representing the generating curve, is controlled by 2architectural parameters and has 40 degrees of freedom.

Moreover, the total number of variables controlling the sec-tions can be reduced with observer functions described in Sec-tion 3.3.

An illustration of the deformation of a sail boat section isshown in Fig.7(b). The three parameters of the section are de-scribed by Fig.7(a). An observer function describes the reparti-tion of the parameters along the sections, but only one sectionof the skeleton is considered in this example, made of 8 controlpoints. The aim of the deformation is to reduce the value ofthe Radius parameters, while Width and Length are fixed. Con-sidering this objective, the extremal points of the sections arefixed. The problem solved has 12 degrees of freedom and con-verges in 15 iterations, with 20 iterations of SQP each.

With a four-cores HP Probook-450 with a Intel R© CoreTM

i7-4702MQ CPU 2.20GHZ, RAM 8.00 GB, the total time toperform this section deformation is 1.2 seconds.

5. Surface reconstruction

The optimisation method outputs deformed sections and gen-erating curves, corresponding to the skeleton of a new shape.To evaluate the shape’s performances with a CFD solver, wefirst need to reconstruct the 3D surface wrapping the deformedskeleton.

Building a new surface allows to obtain a cleaned-up modelfor the meshing tool. The quality of the obtained surfaces is en-sured by the high precision of the implemented fitting processand the control of continuity between patches.

Classical techniques such as lofting [7] are a relevant choicefor objects that can be represented with only one surface suchas foils.

For more complex objects, multi-patch surfaces are required.In such cases, a particular attention has to be given to the con-tinuity between them: for our application, patches have to beat least C1. We developed a technique based on form finding[29] to reconstruct suitable surfaces. We expose this techniquein the following paragraphs.

5.1. Problem setting

Given the section and generating curves of the new deformedskeleton, we construct a surface which contains these curvesand satisfies tangency conditions on the boundaries of the sur-face.

(a) Ship section parameters

(b) Deformation of the curvature radius of a sail boat section

Figure 7: Ship section parameters and deformation

Like for the skeleton deformation, we compute the surfaceby solving an optimisation problem, where the control pointsof the surface ci j are the unknowns. To define this optimisationproblem, we discretize the problem by sampling the curves. Weobtain a point set on which we will wrap the surface. The sur-face is computed by fitting techniques [30], taking into accountsmoothing and tangency constraints on the boundary of the sur-face to ensure the continuity between adjacent surfaces in thegeometric model. As explained in [29], the constraints we useare quadratic in the control point coordinates ci. We describethem in the following paragraphs.

First, we define for each point of the point set a coordinatemapping:

R3 −→ [0, 1] × [0, 1]Pl −→ (ul, vl), l = 0, ...,NP

The mapping defines for each point Pl of the point set, theparameters ul, vl of the surface σ where σ(ul, vl) = Pl will beverified approximately.

Surface fittingThis constraint ensure that the surface σ passes thought thepoints Pl:

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E f itting :∑

l

‖σ(ul, vl) − Pl‖2 = 0, l = 0, ...,NP (5)

with σ(u, v) =

n∑i=0

m∑j=0

ci jBi(u)B j(v)

Tangency constraint with fix parts of the objectLet nl be the normal at Pl of the fix surface adjacent to a sur-face we want to reconstruct. In the u direction, the continuityconstraint is expressed by:

E f ixT : < σu(ul, vl) · nl >2= 0, l = 0, ...,NP (6)

where σu(u, v) =

n∑i=0

m∑j=0

ci jB′i(u)B j(v)

We have similar constraints in the v direction.

Tangency constraint with mobile partsAt the Nn points Pl on the frontier with other reconstructed sur-faces, the values of the normals nk of both surfaces are newunknowns satisfying an equality constraints. In the u direction,the continuity constraint is expressed by:{

EmobileT1 : < σ1 u(u1,l, v1,l) · nk >2= 0

EmobileT2 : < σ2 u(u2,l, v2,l) · nk >2= 0 (7)

l = 0, ...,NP, k = 0, ...,Nn (and similarly for the constraints inthe v direction).

Moreover, the normal vector nk must satisfy:

Emobile normals : < nk · nk >2= 1, k = 0, ...,Nn (8)

Notice that these constraints require an initial value of σ1, σ2and nk.

RegularizationA regularization energy term can also be introduced for the sur-faces, to improve the “fairness” of the surface. It is a quadraticfunction of the unknowns control coefficients ci, j, similar to theregularization term for curves used in Section 4.1. We do notdetail it here (see for instance [29]).

5.2. Numerical solutionLet us consider x as the vector containing the unknown of

the system, in other words the surfaces control points ci j andthe normals nk at the frontier with two reconstructed surfaces.The surface is constructed so that the total energy is minimized:

Etotal = E f itting+E f ixT +EmobileT1+EmobileT2+Emobile normals (9)

A dedicated algorithm is used to compute a value of x, forwhich Etotal is less than a threshold. Let us describe it briefly.

The general form of quadratic constraints that we treat is:

ϕi(x) =12

xT Hix + bTi x + ci = 0, i = 1, ...,N (10)

where Hi is a symmetric matrix, bi is a vector and ci a constant.Some of the constraints that we use are not quadratic e.g. thecontinuity between patches. In such cases we use a geometri-cally meaningful linearization, e.g. expressing the constraint ina quadratic form using the normal of the surface.

Given the definition of the quadratic constraints in Eq. (10)and a value x = xn at iteration n, we can linearize ϕi(x) using:

x = xn + δx

ϕi(x) ≈ ϕi(xn) + ∇ϕi(xn)T (x − xn) = 0, i = 1, ...,N

where ∇ϕi(xn) = Hixn + bTi . We can rewrite this linearization in

the following matrix form:∇ϕ1(xn)T

...∇ϕN(xn)T

· x =

∇ϕ1(xn)T · xn − ϕ1(x)

...∇ϕN(xn)T · xn − ϕN(x)

,that is, a linear system of the form Hn ·x = rn, whose solution isthe next point xn+1. We solve this system iteratively until a fixpoint is reached.

This technique is able to reconstruct efficiently and accu-rately surfaces, which contain the skeleton curves and satisfytangency constraints on the boundary.

We illustrate it with the reconstruction of the surface of asail boat hull, in Fig.8 , and on the bulbous bow of a fishingtrawler, in Fig.9.

For the sail boat hull example, we choose to reconstruct themiddle part of the hull with two surfaces. Each surface has to besmoothly connected to a fix part of the hull (the transom or thestem) and to the other middle surface. The algorithm convergesin 3 iterations, and the resulting surfaces satisfy the tangencyconstraints at the three junction curves.

Figure 8: Patch of surfaces reconstructed of a sail boat hull

For the bulbous bow example, the bulb is reconstructed withtwo surfaces. The first one is connected to the fixed hull and theother is connected to the first surface. The algorithm convergesin 4 iterations, and the resulting surfaces satisfy the tangencyconstraints at the four junction curves.

6. Applications

In this section we present two different applications of theparametric modeller, one on the foil of an AC45 and one onthe bulbous bow of a fishing trawler. In both cases, we aimto increase a performance criterion with shape variations. Theparametric modeller is automatized and linked to a flow solver.A specific flow solver is used for each application: potential

8

Figure 9: Patch of surfaces reconstructed of bulbous bow

flow solver for the foil and RANSE for the bulbous bow.

Timings in the experimentations refer to the same hardwareconfiguration as described in section 4.2.

6.1. Application on a sail boat’s foil: AC45

In the recent years, new high-speed boats were developedusing foils. The purpose of a foil is to lift the hull of the boatabove water surface. The hull resistance (friction and wavemaking drag) is decreasing, allowing to reach very high speeds.

For sailing yachts, the foils are built as an ”L” shape witha vertical part countering the sails forces, and a horizontal partsupporting the yacht weight.

While sailing, the foil allows the yacht to fly as shown inFig.10. However, to maintain this flying state, the stability ofthe foil is a critical aspect for both security and performance.

Designers have to manage numerous parameters in order toproduce a foil with a low drag, but high stability.

We consider here the AC45 foil. This type of foil is ”one-design” meaning that its shape is the same for all AC45 boats.

For this application, we aim to optimise the shape of theAC45 foil in order to decrease its total drag while keeping sta-bility and the ease of use as high as possible. The foil per-formances are computed with the potential flow solver ARA-VANTI.

The AC45 foil is currently used by the Groupama TeamFrance sailing team for the 35th America’s Cup. An illustrationof the sail boat flying thanks to the foil is shown in Fig.10, onefoil in the water (right) and the other one visible in the retractedposition (left).

6.1.1. Simulation with ARAVANTIARAVANTI, the flow code used in the present study is de-

veloped and commercialized by the company K-Epsilon. ARA-VANTI is a coupled fluid-structure solver, with a finite elementmethod for solving the structure and multiple different methodsfor the fluids (e.g. vortex line method, particle method, panelmethod, etc.) [31, 32].

The method used here is a vortex line method with solvedwake. ARAVANTI is coupled to XFOIL in order to incorporatethe flow behaviour such as laminar transition, and stall.

Figure 10: Illustration of the AC45 on the Groupama Team France sail boat,Credit: R© Eloi Stichelbaut / Groupama Team France

The foil is represented with a finite number of elements, i.e.airfoil sections given by the skeleton. For each element a localvelocity, a local Reynolds number and a local angle of attackis computed. Each element has an associated XFOIL databasecontaining the lift and drag of the section for a given range ofangles of attack (usually between −5◦ and 20◦).

ARAVANTI use this database to find the lift of each elementof the foil according to its current local angle of attack. Then thelift is converted to a local vorticity. The wake is imposed withthe computed gradient of vorticity then solved. These steps arerepeated until convergence thanks to a direct iterative method,which is able to find a stationary solution.

For this type of application, ARAVANTI does not need to belinked to any CAD software. As inputs, it requires a 2D pointcloud description of the sections, similar to the XFOIL input,and a text file indicating the 3D position of each section withpoints and quaternions. These files are generated automaticallyby the parametric modeller.

In our specific case for AC45 foil study, only the underwa-ter part of the foil is simulated. The influence of the free sur-face is taken into account with an anti-symmetry plane model.This model is a satisfying approximation for high speed. As[33] suggests, with a Froude number greater than 1, an infiniteFroude number free-surface condition can be used. In our case,the Froude number is around 5.45.

We illustrate in Fig.11 the wake computed with ARAVANTIand the vortex lines. The vortex line is located at 25% of the aftof the leading edge along the foil. From the vorticity repartitioncolormap, we see that the parts of the AC45 which generatemost of the force allowing to lift up the boat are the knee andthe tip.

The reference frame is defined as follows: X is in the oppo-site direction of the flow, Z is in the vertical direction (orientedupwards) and Y is horizontal, perpendicular to X.

9

Figure 11: Illustration of the wake and vortex line on the AC45

6.1.2. Proposed performances criteriaWe choose to define the foil performances with three criteria

computed with ARAVANTI.

1. The total drag Fx of the foil in the reference frame. Alow drag increases the total performance and speed of theboat.

2. A stability criterion, represented by ∂Fz∂z , where Fz is the

total force in the z direction of the foil. The aim of thiscriterion is to ensure that the boat will stay at a fixed zheight thanks to a self adjusting Fz balancing the verticalmovements of the foil.

3. A stability and usage criterion, represented by ∂rake∂V , where

the rake is the angle of incidence of the foil in the Y ro-tation, and V is the boat speed. The rake is a parameterthat the crew have to adjust while sailing to modify thevertical forces Fz. Thus a foil shape where this param-eter does not change a lot when the speed is varying isvaluable.

Computations are performed with a fixed Fy given as theopposite force to balance the force applied by the sails on thehull. Fz is also fixed to counter the weight of the hull and beable to lift it up. The speed of the yacht is first set to 22 knots.ARAVANTI solves for the leeway and rake angles of the foil,until computed forces converge to the imposed forces.

Fx is computed during the simulation, and we aim to de-crease it as much as possible. In the reference frame we used,Fx is oriented along the negative x direction. Thus, the sign ofFx will be negative, but we can consider the absolute value tocompare the foil performance.

To compute the second criterion, we estimate ∂Fz∂z with fi-

nite differences. We vary the foil displacement by a small ∆zand compare the computed Fz. To be stable, the foil has to gen-erate a Fz opposed to the direction of the displacement. Thusthe ratio ∂Fz

∂z has to be negative and as large as possible.For example, if the boat is riding too high above the water sur-face, the foil force Fz has to decrease in order to make the wholesystem lower.

We use the same process for the third criterion, ∂rake∂V , by

solving the rake angle for a small speed variation ∆V . Here,

the rake has to increase as little as possible when the speed in-creases. Thus the ratio ∂rake

∂V has to be positive and as small aspossible.

For both cases, we ensure that the finite difference is a satis-fying approximation by choosing appropriate steps ∆z and ∆V .

The aim of our study is to reduce the total drag of the AC45as much as possible while keeping stability criteria as large aspossible.

6.1.3. Proposed deformationsWe identified the most relevant parameters that influence a

foil performances as the tip length, the angle between the shaftand the tip and the cant angle, illustrated in Fig.12. Here, weconsider the cant angle as a shape parameter and not as controlparameter of sailing.

To generate a new CAD from the original CAD model, ourtools takes on average 12 seconds to build the skeleton, 5.1 sec-ond for the generating curve deformation and 5 seconds for thesection curve deformation. In our case, we perform only de-formation of the generating curve. Moreover there is no needto build a new surface around the skeleton, as ARAVANTI doesnot require a continuous surface as an input. A set of points dis-tributed on the section curves of the skeleton is sufficient. Theskeleton we used on the AC45 is illustrate in Fig.2(a).

Figure 12: Foil shape parameters

The variations of the parameters are distributed in a param-eter space defined in Tab.1, and illustrated in Fig.13.

Tip length Angle CantInitial value 1.37m 77.24◦ 2.42◦

Min variation −30% −30% −313.7%(= 0.96m) (= 54.1◦) (= −5.2◦)

Max variation +40% +20% +727.2%(= 1.92m) (= 92.65◦) (= 20◦)

Table 1: Limits of parameters domain

To sample the parameter space, we use a Latin Hypercubedistribution [34]. Our choice is based on the future use of op-timisation algorithms such as EGO, that are often initialized

10

Figure 13: Shape variation of the foil in the paramter space

with such parameter space values distributions as they are welladapted for response surface methods [35].

6.1.4. ResultsWe used a Latin Hypercube distribution with 20 points to

sample the parameter space described in Tab.1. For each set ofparameters, we build a new corresponding foil with our para-metric modeller and we evaluate automatically the value of the3 criteria, Fx; ∂Fz

∂z ; ∂rake∂V , with ARAVANTI.

As our aim is to reduce the total drag as much as possiblewhile keeping stability criterion as large as possible, the opti-mal solution is located on a Pareto front. We represented thePareto fronts of the drag with each stability criterion in Fig.14.The Pareto front is defined with few points at this stage of thestudy. More points will be added while performing the auto-matic shape optimisation process. In a future work, the perfor-mance of the foils located on the current Pareto front will beimproved by including an adapted optimisation algorithm.

We named the foils closest to the Pareto fronts (A,B,C,D),Foil A being the one with the least drag and worst stability, FoilD being the one with the most drag, but the best stability andFoils B & C being in between. Even if Foil A has the worststability of the Pareto front, it is still better than the originalAC45. The other criteria vary around the original values.

Note that the foils A, B, C and D refer to the same shapeson both Pareto fronts ∂Fz

∂z vs Fx and ∂rake∂V vs Fx.

We detail the points on the Pareto fronts in Tab.2, with theinitial AC45 results for comparison. We illustrate the results inFig.15.

The two shape variations Foil A and Foil B are rather dif-ferent for the tip length and angle values. We can deduce a linkbetween these two parameters that leads to more efficient foils,either a short tip with a great angle or a long tip with a smallangle. Both cases suggest to increase the cant angle.

The two extreme shapes in the Pareto front ∂Fz∂z vs Fx, Foil

A and D, show a very different behaviour of the foil accordingto the parameters, illustrated in Fig.16 where we see the vortic-ity distribution along the foil. In the case of Foil A (Fig.16(a))the vorticity is uniformly distributed on the shaft, knee and tip.

Figure 14: Latin hypercube distribution of foils shape

# % Tip length % Angle % Cant Total drag ∂Fz∂z

∂rake∂Vvariation variation variation (|Fx|) in N

AC45 - - - 1077 −423 1.045

A −8.62% +18.58% +650.53% 983 −1495 0.825

B +34.02% +9.74% +416.79% 983 −2863 0.710

C +31.56% −5.96% +591.87% 1122 −6392 0.872

D +11.88% −29.63% +721.38% 1327 −7240 1.271

Table 2: Parameters and criteria values of points on both Pareto fronts

Whereas for Foil D (Fig.16(b)), the vorticity in essentially lo-cated on the shaft, thus the lifting force is principally generatedfrom this part.

To conclude, the behaviours we observed of the differentfoils match expected results, and some tendencies are well knownby designers.

A further study will include the sinkage as well as shapeparameters for the sections. We will also take into account themoment of the boat about the x direction Mx. The momenthas an influence on the predicted performance of the foil, andespecially the value of cant angle can be affected in order to finda configuration that counters Mx.

Also, an optimisation algorithm will be integrated in theloop, helping to determine with certitude the best tendency ofparameter values.

6.2. Application for a bulbous bow

We present an application of our parametric modeller fordeforming a fishing trawler bulbous bow.

11

(a) Foil A vs AC45 (b) Foil B vs AC45

(c) Foil C vs AC45 (d) Foil D vs AC45

Figure 15: Shape variations on Pareto fronts

The original trawler was designed without a bulbous bow.We aim to reduce the total drag of the hull by adding a bulbousbow.

An initial bulb was designed by a naval architect, then wepropose to vary three parameters to control the shape: the anglethe length and the width at mid-bow of the bulb.

To generate a new CAD from the original CAD model, ourtools takes on average 27.6 seconds to build the skeleton, an av-erage of 14.1 seconds to perform deformations, and 20 secondsto reconstruct the surface.

RANS (Reynolds-Averaged NavierStokes equations) simu-lation being more complex to set-up, the link with the paramet-ric modeller was not fully automatised.

6.2.1. Simulation with FINETM/MarineTo generate non-conformal, fully hexahedral, unstructured

meshes for complex arbitrary geometries, we use HEXPRESSTM

from Numeca International. The advanced smoothing capabil-ity provides high-quality boundary layers insertion [36]. Thesoftware HEXPRESSTM creates a closed water-tight triangular-ized volume, embedding the ship hull, then a body-fitted com-putational grid is built. One of the meshes used in our simula-tions is shown in Fig.17.

The grid generation process requires a clean and closed ge-ometries to provide robust meshes. Thanks to the shape con-sistency control and the smooth reconstruction of surfaces, the

(a) Foil A

(b) Foil D

Figure 16: Wake of Foils A and D, front view

modeller generates shapes which are well-adapted to these re-quirements and which allow to produce high-quality meshes forcomputations.

During the computation, automatic mesh refinement has beenused. Automatic, adaptive mesh refinement is a technique foroptimising the grid in the simulation, by adapting the grid to theflow as it develops during the simulation to increase the preci-sion locally. This is done by locally dividing cells into smallercells, or if necessary, by merging small cells back into largercells in order to undo earlier refinement. During the computa-tion, the number of cells increases from 1.9 to approximativelyto 2.2 million cells, for a half hull mesh. Fig.17 shows the meshrefinement around the hull and the free surface at the end of thecomputation.

We use the flow solver ISIS-CFD, available as a part of theFINETM/Marine computing suite. It is an incompressible, un-steady Reynolds-averaged Navier-Stokes (RANS) solver [37,38]. For the turbulent flow, additional transport equations forthe modeled variables are discretized and solved. The two-equation k-ω SST linear eddy-viscosity model of Menter isused for turbulence modeling. The solver is based on the finitevolume method to build the spatial discretisation of the trans-port equations.

12

Figure 17: View of the mesh around the hull with free free surface deformation

The unstructured discretisation is face-based, which means thatcells with an arbitrary number of faces are accepted. This makesthe solver ideal for adaptive grid refinement, as it can performcomputations on locally refined grids without any modification.Free-surface flow is simulated with a volume of fluid approach:the water surface is captured with a conservation equation forthe volume fraction of water, discretised with specific compres-sive discretisation schemes, [38]. The vessel’s dynamic trimand sinkage are resolved during the simulation.

The studied trawler has a waterline length of 22.35 metresand a displacement of 150 metric tons. Simulations are doneat a speed of 13 knots (6.688m/s), corresponding to a a Froudenumber of 0.4517. Trim and sinkage are solved. The hull for-ward motion is imposed by accelerating it to its final speed witha 1

4 sinusoidal ramp. Fluid characteristics are shown in Tab.3.

ρ (kg/m3) µ (Pa.s)Water 1026.02 0.00122

Air 1.2 1.85 ∗ 10−5

Table 3: Fluid characteristics

6.2.2. Proposed deformationsThe skeleton used for the bulbous bow is illustrate in Fig.2(b).We propose to vary three parameters to control the shape:

the angle, the length, and the width at mid-bow of the bulb.Fig.18 illustrates variations of bulbous bow shape according tothe parameters.

(a) Length variation (b) Angle variation (c) Width variation

Figure 18: Bulbous bow shape variations

Parameters such as sectional area, vertical position of thecentroid, type of sections (delta, oval, nabla) are also importantparameters for the shape of the bulbous bow. These parametersare being integrated into the parametric modeller. The currentstudy focuses on the three parameters (angle, length and width),and will be extended in a future work with new parameters.

The variations are distributed in a parameter space definedin Tab.4, according to limits given by architectural criteria.

The initial bulb being quite short, we assumed that shapeswith a lower length than 1.86m will not positively influencethe drag, likewise we restricted the bulb to not be longer thanthe extremity of the upper bow. For the angle, we noticed thatwhen the length of the bow is increased, keeping the originalvalue will cause the bulb to pierce the free surface, again thisconfiguration is unwanted.

Length Angle WidthInitial value 1.61m 31.52◦ 0.83m

Min variation +15% −25% −20%(= 1.86m) (= 23.64◦) (= 0.66m)

Max variation +90% 0% +20%(= 3.07m) (= 31.52◦) (0.99m)

Table 4: Limits of parameters domain

As for the application to the foil, we use a Latin Hypercubedistribution to sample the parameter space in order to preparea relevant dataset for the future use of optimisation algorithmssuch as EGO.

6.2.3. ResultsWe used a Latin Hypercube distribution with 20 values. We

present in Tab.5 the results of the original hull without bulb, thehull with the initial bulb and the best variation obtained fromthe parameters variation. The best variation is not a fully opti-mised bulbous bow, as no optimisation algorithm has yet beenused for this application. It is planned to use EGO [35] on thedistribution obtained through this study as a future work.

The best drag reduction is obtained with the following pa-rameter values : Length: +58.70% (= 2.56m) ; Angle: −19.81%(= 25.28◦) ; Width: +9.99% (= 0.66m). The total drag Fx iscomposed of the friction resistance FFric,x component and thepressure resistance FPres,x component. The friction resistanceis obtained by integrating the tangential component of stressesover the wetted surface of the hull, in the direction of motion,and the pressure resistance is obtained by integrating the nor-mal component, in the direction of motion. Tab.5 presents thedifferent components of resistance computed by ISIS-CFD anda comparison with ITTC-57 friction resistance computed withdynamic wetted surface. There is no correlation allowance usedas the computation is performed at full scale.

In other terms, the best variation represents an improvementof 3.64% from the first bulb design. The pressure resistance rep-resents an important part of the total drag of the original hull.

13

Original hullInitial bulb Best variation

(without bulb)Static wetted

183.6 190.1 193.7surface in m2

Dynamic wetted195.4 200.7 204.2

surface in m2

Total drag Fx 79910 73740 71054in N

Pressure resistance FPres,x 70599 63852 60970in N

Friction resistance FFric,x 9311 9887 10083in N

ITTC-57 friction resistance computed9042 9140 9445

with dynamic wetted surface in N% reduction

- 7.72% 11.08%of drag Fx

Table 5: Drag results and variations on the bulbous bow

We can observe the positive impact of the bulbous bow on thiscomponent. It reduces the pressure resistance by modifying theshape of the bow wave (see Fig.19). The friction resistance in-creases with the wetted surface. Despite that the static/dynamicwetted surface increases due to the additional surface of the bul-bous bow, the total resistance decreases thanks to its effect onthe flow.

Trim affects the drag by modifying the flow around the shiphull. The volume distribution of the bulbous bow is not pre-served during the shape deformation, but the total displacementand location of the center of gravity are kept identical for eachdesign.

Tab.6 shows the evolution of the total trim and sinkage ofthe three hulls. The total trim (respectively sinkage) is the sumof the hydrostatic the dynamic trim (respectively sinkage). Thevariation of total trim is relativity small between the differentdesigns. As trim of the original hull and best variation are simi-lar, the effect on the drag is mostly produced by the modificationof the shape of the bulbous bow.

Total trim in deg Total sinkage in mOriginal hull 1.656 −0.257(without bulb)Initial bulb 1.495 −0.264

Best variation 1.728 −0.225

Table 6: Trim and sinkage results

Fig.19 illustrates the free surface elevation of the two de-signs, Initial bulb and Best variation, and Fig.20 show the wavepatterns.

The sampling we performed with the Latin Hypercube isrepresented graphically with a response surface method, illus-trated in Fig.21. Figure 21(a) represents cutting planes of thedesign space, showing two main local minima. In Figure 21(b),we show iso-values of the total drag Fx. We can identify a re-gion where the objective function is predicted to be smaller thanin the other parts of parameter domain.

Further investigations may lead to finding better drag reduc-

(a) Free surface elevation for the Initial bulb

(b) Free surface elevation for the Best variation

Figure 19: Free surface elevation

tion results by using an adapted optimisation algorithm basedon Kriging such as EGO to find minima using the model builtfrom the response surface.

7. Conclusion and future work

This paper presents a method for parametrizing and deform-ing different type of shapes with a skeleton-based approach.The methodology we develop reduces the number of degrees offreedom thanks to observer functions described with B-Splinesand provides a fine control of the geometry in terms of archi-tectural parameters. Our tool can handle any shape that can bedescribed with the skeleton-based parametrization.

Our parametric modeller allows to explore the domain ofpossible shapes in an efficient way, and allows to determine im-provements of the design that are architecturally relevant.

As shown by the experiments, we are able to improve thehydrodynamic performances of a AC45 foil and a bulbous bow,with a few number of parameters.

Further work will focus on handling more complex geome-tries with the skeleton representation. Section curves with mul-tiple components and branching curves will be considered.

14

Figure 20: Wave generation of the initial bulb (upper part) and the best variation(lower part) designs

(a) Cutting planes of the response surface

(b) Iso values of the total drag Fx in the response surface

Figure 21: View of the response surface

We will also develop the link with optimisation algorithmsolvers. A fully automatised optimisation loop will be devel-oped. Sensitivity of the simulation results to parameters will betaken into account in order to reduce the degrees of freedom asmuch as possible.

Acknowledgements: The project was achieved with the finan-cial support of ANRT (Association Nationale de la Rechercheet de la Technologie).

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